Algebra II WiSe2022-2023

0. Introduction: a glimpse of Algebraic Geometry

I. Rings, ideals and modules

I.2. The spectrum of a ring

I.3. More on PID and UFD

I.4. Localization

I.5. Radicals

I.6. Modules

I.7. Tensor products

I.8. Change of rings

I.9. Nakayama's lemma

II. Finiteness conditions

II.1. Some homological algebra}

II.2. Finitely presented modules}

II.3. Noetherian rings and modules

II.4. Artinian rings and modules

II.5. The structure of Artinian rings

II.6. Connected components

II.7. Irreducible components

II.8. Primary ideals

II.9. Primary decomposition

III. Integral extensions and the Nullstellensatz

III.1. Finite and integral extensions

III.2. The integral closure

III.3. Going up and going down

III.4. Noether normalization

III.5. Hilbert's Nullstellensatz

IV. Dimension

IV.1. Krull dimension

IV.2. Equidimensional rings}

IV.3. Dimension of affine varieties

IV.4. Transcendence degree